The scaled-invariant Planckian metal and quantum criticality in Ce1−xNdxCoIn5

The mysterious Planckian metal state, showing perfect T-linear resistivity associated with universal scattering rate, 1/τ = αkBT/ℏ with α ~ 1, has been observed in the normal state of various strongly correlated superconductors close to a quantum critical point. However, its microscopic origin and link to quantum criticality remains an outstanding open problem. Here, we observe quantum-critical T/B-scaling of the Planckian metal state in resistivity and heat capacity of heavy-electron superconductor Ce1−xNdxCoIn5 in magnetic fields near the edge of antiferromagnetism at the critical doping xc ~ 0.03. We present clear experimental evidences of Kondo hybridization being quantum critical at xc. We provide a generic microscopic mechanism to qualitatively account for this quantum critical Planckian state within the quasi-two dimensional Kondo-Heisenberg lattice model near Kondo breakdown transition. We find α is a non-universal constant and depends inversely on the square of Kondo hybridization strength.


Supplementary Note 2: Estimation of carrier concentration n and α coefficient
In this section, we provide derivation of the relevant equations for carrier concentration n based on the quantum oscillation measurements. We will further use those equations to reproduce n and the Planckian coefficients α shown in Table 1 of the main text. We start from the formula of quantum oscillation frequency F, given by with S F being the extremal cross-sectional area of the Fermi surface. For simplicity, we assume a circular cross-section of Fermi surface here, hence S F = πk 2 F with k F being the "averaged" Fermi wave vector of the circular Fermi surface. This links the dHvA frequency and the "average" Fermi wave vector of the Fermi surface by F = (h/2π)k 2 F 2e . Here, we can make a link of F and the carrier concentration n through k F . While considering the effective dimensionality of critical modes, the carrier concentration takes the following form 1 : where d b and d c are the lattice constants of unit cell along the b and c axes. Note that, for the 2d case of Eq. (S2), we assume the system has a strong anisotropy along the c direction while it remains isotropic in the a-b plane.
As an example, we provide detailed derivation of carrier concentration for the 2d case shown in Eq. (S2) and then generalize this derivation to the case with fractional quasi-2d dimension.
Assume the critical modes occur on the isotropic ab-plane. The total number of states can be expressed as where N ab = be generalized for the case of arbitrary dimensional critical modes with fractional quasi-2d dimensionality embedded in a 3d lattice.
The total number of states per spin for an isotropic ddimensional system is given by where In the above equation, Γ (x) denotes the Γ function. For a general d-dimensional critical modes embedded in a 3d lattice, the total number of states reads where the prefactor 2 comes from the spin degrees of freedom.
Here, we assume that the quasi-2d critical modes mostly arise from the ab-plane. From Eq. (S4), we have The total number of states is then given by giving rise to the carrier concentration . (S10)

2/4
Using the relation of k F and F, we obtain the expression of carrier concentration for arbitrary d-dimensional critical modes, (S11) When taking d = 2, the above expression of n goes back the 2d case in Eq. (S2).

Supplementary Note 3:
Estimating the Planckian coefficients α for Ce 1−x Nd x CoIn 5 Below, we estimate the carrier concentration n and Planckian coefficients α shown in Table 1 of the main text for Ce 1−x Nd x CoIn 5 with x = 0, 0.02, 0.05, and 0.1 using the dHvA frequency F and effective mass m ⋆ in Ref. 2 (for αband) and in Refs. 1, 3 (for β -band).
• For x x x = = = 0 0 0. . .0 0 02 2 2. The average dHvA frequency for the αband is F = 4.89kT while m ⋆ = 11.7m 0 is its average effective mass. Angular dependence of the dHvA frequencies indicates a 2d Fermi surface of the α-band for x = 0.02 2 . Following the similar approach, the carrier concentration of the α-band is calculated as n = 0.32 × 10 28 m −3 (for α-band). (S16) Here, we assume that the carrier density and the relevant band parameters as well as the effective dimension of the β -band do not significantly altered while doping 2% of Nd, indicating that n = 0.63 × 10 28 m −3 and m ⋆ = 100m 0 for the β -band here. The α-coefficients for the α-and β -band are thus estimated giving the total Planckian coefficient α = 0.72 for x = 0.02. The gradient of the linear-T resistivity A 1 = 1.0µΩ · cm/K is used in this case.
• For x x x = = = 0 0 0. . .0 0 05 5 5. The fundamental band parameters of the α-band for x = 0.05 is F = 4.88kT and m ⋆ = 9.15m 0 . Angular dependence of the dHvA frequencies indicates a 2d-to-3d dimensional crossover of Fermi surface of the α-band at x = 0.05 2 . Accompanying with the prediction of a QCP at x c = 0.03 and the theoretical studies on that QCP, we treat the dimensionality for x = 0.05 to be d = 2.45. Using Eq. (S11), the carrier concentration of α-band is estimated as n = 0.26 × 10 28 m −3 (for α-band). (S18) Likewise, we assume the band parameters of the β -band also remains the same for x = 0.05, thus m ⋆ = 100m 0 and F = 9.75kT. The carrier concentration of the β -band with d = 2.45 is found to be n = 0.6 × 10 28 m −3 (for β -band). (S19) Using A 1 = 1.17µΩ · cm/K for x = 0.05, the αcoefficients for the α-and β -band can be straightforwardly calculated as α = 0.7 (for α-band), giving the total Planckian coefficient α = 0.85.
• For x x x = = = 0 0 0. . .1 1 1. The fundamental band parameters of the α-band for x = 0.1 is F = 4.41kT and m ⋆ = 7m 0 . We treat the effective dimensionality of Fermi surface of the α-band to be three-dimensional as this compound at x = 0.1 is deep inside the AF state 4 . Using Eq. (S11) and Similarly, we assume that m ⋆ = 100m 0 and F = 9.75kT are also applicable for the β -band for x = 0.1 here. The carrier concentration of the β -band with d = 3 is found to be n = 0.55 × 10 28 m −3 (for β -band). (S22) Using A 1 = 1.49µΩ · cm/K for x = 0.1, the αcoefficients for the α-and β -band can be straightforwardly estimated as α = 0.77 (for α-band), α = 0.19 (for β -band), giving the total Planckian coefficient α = 0.96.